Sen. Lindsey Graham, the sole Republican who was working on a bill to legalize illegal immigrants, in effect put the bill on the shelf on Tuesday, saying a debate now would destroy any prospects for passage and saying the issue needs to wait until 2012.

The remarks likely signal the end of any serious chance for broad immigration legislation to pass this year, since Mr. Graham, South Carolina Republican, was the best hope for a partnership with President Obama and Democrats who want to write a bill.

“It is impossible for me and any other serious Democrat to get this body to move forward until we prove to the American people we can secure our borders,” Mr. Graham told Homeland Security Secretary Janet Napolitano, who was testifying at a Senate Judiciary Committee hearing.

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“I believe we can do it by 2012 if we’re smart,” he said.

Ms. Napolitano, a former governor of Arizona, disagreed with Mr. Graham’s evaluation of border security. She said she knows the southwest border as well as anyone and said by every measure Congress has laid out, the border is more secure: Fewer illegal immigrants are being apprehended, and more fencing and infrastructure have been deployed.

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But under close questioning by Mr. Graham, Ms. Napolitano could not say whether she would declare the border secure if she were still the governor of Arizona. She called it an “unfair question.”